Two fixed points $A$ and $B$ and a given real number $K$,find the locus of point $M$ for each of which $$2MA^2+3MB^2=K$$
My attempt:
By trial and error I could figure out that at $K=\frac{6}{5}AB^2$ ,the desired locus of point M is a single point which divides the join of $A$ and $B$ in ratio of $3:2$.However i couldnot find out the generalised locus for any value of $K$.Please help me out with this problem.Thanks
Hint:
For $A=(x_A,y_A)$ and $B=(x_B,y_B)$ the locus is the set of points $M=(x,y)$ such that: $$ 2\left[(x-x_A)^2+(y-y_A)^2 \right]+3\left[(x-x_B)^2+(y-y_B)^2 \right]=K $$
can you do from this?